Wrong behavior when I try to apply the UV grid texture to my object
In the first screenshot, it seems that the penguin has no material, unlike in the last screenshot where the penguin has 3 materials assigned (and there it shows the texture)..?



Could it be just this?

Edit:
Following you request in comments/chat, I did a test:

1.I  downloaded your file, saved on my desktop


I created a new default file in blender
I appended just the penguin mesh from your file (I found it's named "untitled" in your file, under "mesh" folder.
Then I selected my default cube object, and switched to the "object data" in its properties panel
From the "browse mesh data to be linked" selection tool, I selected the previously imported penguin mesh, named "untitled".
Now my "cube" object has a penguin shape. Your UV grid also gets imported.
Setting viewport shading to "textured" shows the textured penguin!




So, I think the object in your file was corrupted, but the mesh is fine.

You can correct your file in this way, I think.

Btw, your mesh has arms normals inverted, select all mesh vertices, and fix them (CTRLN).

This is my file:


Hth.